Chapter 2
When x = a is a zero of a polynomial function f, the following three statements are true: (a) x = a is a _________ of the polynomial function f(x) = 0. (b) _________ is a factor of th epolunomial f(x). (c) (a,0) is an _________ of the graph of f.
(a) Solution (b) (x - a) (c) X-intercept
The graph of a quadratic function is symmetric about its _________.
Axis of symmetry
The numbers a + bi and a - bi are called _________ _________, and their product is a real number a squared + b sqaured.
Complex conjugates
If a + bi is a complex zero of a polynomial with real coefficients, then so is its _________, a - bi.
Conjugate
The graphs of the functions are _________, which means that the graphs have no breaks, holes, or gaps.
Continuous
The theorem that can be used to determine the possible real numbers of positive real zeros and negative real zeros of a function is called a _________ _________ of _________.
Descartes' rule of signs
The _________ Theorem states that a polynomial f(x) has a factor (x - k) if and only if f(k) = 0.
Factor
The _________ _________ of _________ states that if f(x) is a polynomial of degree n (n > 0), then f has at least one zero in the complex number system.
Fundamental Theorem of Algebra
When f(x) ---> b as x ---> positive/negative infinity, y = b is a _________ _________ of the graph of f.
Horizontal asymptote
An _________ number has the form a + bi, where a does not equal 0, and b does not equal 0.
Imaginary
In the Division Algorithm, the rational expression f(x)/d(x) is _________ because the degree of f(x) is greater than or equal to the degree of d(x).
Improper
The _________ _________ Theorem states that if a polynomial function such that f(a) does not equal f(b), then, in the interval (a,b), f takes on every value between f(a) and f(b).
Intermediate Value
A quadratic factor that cannot be factored further as a product of linear factors containing real numbers is said to be _________ over the _________.
Irreducible over the reals
To solve a polynomial inequality, find the _________ numbers of the polynomial, and use these numbers to create _________ _________ for the inequality.
Key Test intervals
The _________ _________ _________ is used to determine the left-hand and right-hand behavior of th egraph of a polynomial function.
Leading coefficient test
Every polynomial of degree n > n with real coefficients can be written as the product of _________ and _________ factors with real coefficients, where the _________ have no reaal zeros.
Linear Quadratic Quadratic
The _________ _________ _________ states that if f(x) is a polynomial of degree n (n > 0), then f(x) has prexisely n linear factors. (THE REST)))))))))))
Linear Factorization Theorem
A real number b is a _________ bound for the real zeros of f when no real zeros are less than b, and is a _________ bound when no real zeros are greather than b
Lower Upper
MISSING NUMBER 2
Missing NUMBER 2
A polynomial function of a degree n has at most _________ real zeros and at most _________ turning points.
N N - 1
When the graph of a quadratic opens downward, its leading coefficient is _________ and the vertex of the graph is _________.
Negative Maximum
The formula that relates cost, revenue, and profit is _________.
P = R - C
Linear, constant, and squaring functions are examples of ________ functions.
Polynomial
When the rgaph of a quadratic function opens upward, its leading coefficient is _________ and the vertex of the graph is _________.
Positive Minimum
Between two consecutive zeros, a polynomial must be entirely _________ or entirely _________.
Positive Negative
When a is a positive real number, the _________ _________ root of -a is defined as the square root of -a, which equals the square root of a (i).
Principle square
In the Division Algorithm, the rational expression r(x)/d(x) is _________ because the degree of r(x) is less than the degree of d(x).
Proper
A _________ _________ number has the form a + bi, where a equals 0, but b does not equal 0.
Pure Imaginary
A _________ function is a second-degree polynomial function, and its graph is called a _________.
Quadratic Paraboloa
Functions of the form f(x) = N(x)/D(x), where N(x) and D(x) are polynomials and D(x) is not the zero polynomial, are called _________ _________.
Rational function
The test that gives a list of the possible rational zeros of a polynomial function is the _________ _________ Test.
Rational zero test
A _________ number has the form a + bi, where a does not equal 0, but b does equal 0.
Real
The _________ Theorem states that if a polynomial f(x) is divided by x - k, then the remainder is r = f(k).
Remainder
A factor (x - a) raised to k, k > 1, yields a _________ _________ x = a of _________ k.
Repeated zero Multiplicity
Chapter 2
Section 1
Chapter 2
Section 2
Chapter 2
Section 3
Chapter 2
Section 4
Chapter 2
Section 5
Chapter 2
Section 6
Chapter 2
Section 7
For the rational function f(x) = N(x)/D(x), if the degree of N(x) is exactly one more than the degree of D(x), then the graph of f has a _________ (or oblique) _________
Slant asymptote
The imaginary unit i is defined as i = _________, where i squared = _________.
Square root of -1 -1
A polynomial function is written in _________ form when its terms are written in descending order of exponents from left to right.
Standard
An alternative method to long division of polynomials is called _________ _________, in which the divisor must be of the form x - k.
Synthetic division
When a real zero of a polynomial function is of even multiplicity, the graph of f _________ the x-axis at x = a, and when it is of odd multiplicity, the graph of f _________ the x-axis at x = a.
Touches Crosses
When f(x) ---> positive/negative infinity as x ---> a from the left or right, x = a is a _________ _________ of the graph f.
Vertical asymptote
The key to numbers of a rational expression are its _________ and its _________ _________.
Zeros Undefined values
